Department Colloquium: Roland Bauerschmidt

The central limit theorem of probability theory asserts under very general assumptions that properly renormalised sums of independent and identically random variables converge to a normal distribution. It can be viewed as a global stability result for a fixed point of a dynamical system. The dynamical system is given by sucessive convolution and the fixed point is the normal distribution. Here independence plays an important role by making the dynamical system autonomous. In statistical mechanics, collections of very strongly dependent random variables are at the heart of many problems. The renormalisation group is a grand extension of the dynamical view of the central limit theorem to systems with strong dependence and spatial structure, in which non-trivial phase portraits arise. I will discuss its background and some applications.